Cover Story: The Fascination of Mathematics

The Sound of Mathematics

By Paul Janositz

Olga Holtz was granted the Sofja Kovalevskaja Award in 2006, upon which she left Berkeley and came to work in Berlin.

Yes, she would like to play a role, especially in an opera – Olga Holtz just adores singing. When she is asked what role she would like to play most, Carl Friedrich Gauss, Henri Poincaré or Grigori Perelman, she answers, “Sofja Kovalevskaja.” A logical choice, after all, the Russian pioneer, the first female doctor and professor of mathematics was not only a compatriot and fellow colleague: “There’s an awful lot about her life I understand quite intuitively,” the 34 year-old researcher comments.

Kovalevskaja, who was tutored a hundred and fifty years ago by the great mathematician Karl Weierstrass in Berlin, is the eponym of one of the most valuable German research awards in existence. It is granted by the Alexander von Humboldt Foundation to entice young, cutting-edge researchers from all over the world to come to Germany. Olga Holtz was granted the award in November 2006 and chose Berlin’s Institute of Technology (TU).

Olga HoltzFoto: Doris Spiekermann-Klaas/
Der Tagesspiegel

Perhaps she would have ended up in Berlin, the city of three opera houses, if she had followed her musical proclivities, too. Her home in the Russian city of Chelyabinsk was full of music and mathematics. Her parents are programmers who taught their daughter algorithms, the rules of calculation on which computers function. Music or mathematics? The 15 year-old schoolgirl had to make a difficult decision. She chose arithmetic.

“Music is beautiful, but mathematics are more interesting,” says Olga Holtz. The researcher laughs as she often does during the interview in the TU’s Institute of Mathematics. With her fashionable clothes, long, wavy hair and open expression she completely thwarts the cliché of the brooding mathematician. So you don’t actually have to be solitary, unworldly, unapproachable like Holtz’ brilliant compatriot Grigori Perelman in order to be a successful mathematician. Perelman was the first to prove the Poincaré Conjecture, a conundrum that had remained unsolved for more than a hundred years. In 2006, when Perelman was to be awarded the Fields Medal, one of the most distinguished honours in mathematics, he turned it down and withdrew to the forests around St. Petersburg.

“Perelman looks on mathematics as something purely objective and, at the same time, something beyond the human,” Holtz comments. “But this isn’t the case.” The petite researcher sits at her desk as straight as a ramrod. “It’s people who do mathematics, just like chemistry, physics or biology,” she explains. That is why science could never be perfectly objective.

“Olga Holtz is one in a million,” says Volker Mehrmann, professor of mathematics at the TU. He is her mentor who hosted her back in 2002 when she came to the TU for the first time as a fellow of the Humboldt Foundation. Prior to that, further study and a doctorate had taken her from her home town in the Urals across the world. After her Humboldt-year in Berlin she returned to the USA, to Wisconsin and then to Berkeley. The Californian elite university offered her a professorship. But the Kovalevskaja Award brought Olga Holtz back to Berlin.

“Berlin’s so exciting, it’s a hive of mathematical activity,” she says. Indeed, the discipline is not only flourishing at the three Berlin universities and the Weierstrass and Zuse institutes, the Matheon Research Centre, funded by the German Research Foundation, has also established itself here, and then there is the Berlin Mathematical School, a graduate school in the framework of the German “Excellence Initiative”. “If we are talking mathematics, Berlin is right at the front of the field worldwide,” according to Leibniz award winner Günter Ziegler, President of the German Mathematicians’ Association, Professor at the TU and special reviewer and host for the Humboldt Foundation.

So just the right place for someone who likes trying “to crack the hardest nuts,” as Mehrmann emphasises. During her doctorate at the Uni of Wisconsin Holtz just quickly solved a few “mathematical conjectures”. And she learns languages at a rate of knots, too. Indeed, the researcher speaks almost perfect German. No wonder the mathematician from the TU is proud to have this multi-talent at his institute. Olga Holtz laughs when this is mentioned. “What would they have to offer you to make you stay in Berlin?” The researcher takes her time to reply. “I think it’s too early to talk about that yet,” she says finally. For the time being she is pleased to have got a team together. Building up the six-strong, multinational working group had taken a lot of time and effort which she had had to fit in between trips to California.

Now Olga Holtz wants to get on with the research proper. “Practical theory” is the way she describes her approach. Of course, in mathematics theory always comes first, she explains. First of all, the formulae have to be developed. But she uses the formulae to solve concrete application problems, not just in mathematics but in physics, chemistry and biology, too. Even computers can be speeded up in this way. Holtz’ big advantage here is that she is just as at home with computers as with mathematics. These were the fields she had specialised in at the Department of Computer Sciences in Wisconsin. For this artistically-gifted researcher Berlin is more than just a hive of mathematical activity. She loves its international flair. “I think it’s great to hear Italian, French or Turkish all being spoken in the street,” she says. She enjoys “shopping at the Kaufhaus des Westens, having a double latte at Caras in Prenzlauer Berg, the play of light and shade in the evening light of summer”. Olga Holtz talks about the many theatres, the opera houses, the libraries. “Well, I just enjoy the life here,” she comments

And as she loves singing, she asked Volker Mehrmann about choirs. The only one the mathematician could think of was the Berlin Philharmonic Choir. And in no time, Holtz was a member and now sings with them regularly. She pulls a CD out of her desk drawer. “Our last concert, Beethoven’s Missa Solemnis,” she announces proudly. According to her colleague Mehrmann, she’s a “multi-talent with enormous self-confidence”. An accurate assessment.

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Dr. Paul Janositz, who studied chemistry, works for the knowledge and research section of the “Tagesspiegel”, specialising in scientific, environmental and technical subjects. This piece is a slightly modified version of an article by Paul Janositz that appeared in the “Tagesspiegel” on 7 April 2008.

Numerical Analysis: Solving millions of equations concurrently

Olga Holtz does applied mathematics. Her special field is numerical analysis. She researches into “practical problems for which there aren’t yet any good theories”. The central topics are computation using matrices and eigenvalue problems.

“Of course, results do exist, but no-one knows, for example, exactly how quickly matrix computation can be carried out,” Holtz explains. Completely new theoretical methods have to be developed to do this.

Matrices are special tables with many thousands of columns and rows. Under certain conditions the entries contained therein can be added or multiplied. A characteristic quantity for the solution is the eigenvalue. Matrices can be used to solve complex systems of equations and process huge amounts of data which often accrue in relation to practical problems.

Whether we are dealing with the aerodynamic properties of aircraft, developing new drugs, or scheduling optimum timetables for the underground, millions of equations constantly have to be solved quickly and precisely. Only computers can do it. Speed is decisive for the Internet, too.

In order to be able to do any calculations with this mass of data in the first place, it has to be decomposed as slickly as possible. “Fast matrix multiplication” is Olga Holtz’ speciality. The trick she uses is to tolerate a certain degree of deviation from exact results. She differentiates between “stable” solutions, which she continues to process, and “instable” ones which she discards. And in this way she achieves acceptable results more quickly. “I transform these operations into software,” the mathematician explains. Thanks to her turbo algorithms, computers can deal with enormous amounts of data at record speed.